Question

which expression is equivalent to \\(\frac{(6y^3)^{-2}}{y}\\) for all values of \\(y\\) where the expression is defined? \\(\bigcirc\\ \frac{1}{12y^6}\\) \\(\bigcirc\\ \frac{1}{36y^5}\\) \\(\bigcirc\\ \frac{1}{12y^7}\\) \\(\bigcirc\\ \frac{1}{36y^7}\\)

Explanation:

Step1: Apply the negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\) (or \(\frac{1}{a^{-n}} = a^{n}\)). So, we can rewrite \((6y^{3})^{-2}\) as \(\frac{1}{(6y^{3})^{2}}\).
Now our expression becomes \(\frac{\frac{1}{(6y^{3})^{2}}}{y}\), which is the same as \(\frac{1}{(6y^{3})^{2}\cdot y}\) (because dividing by \(y\) is multiplying by \(\frac{1}{y}\)).

Step2: Simplify the denominator \((6y^{3})^{2}\)

Using the power of a product rule \((ab)^{n}=a^{n}b^{n}\), we have \((6y^{3})^{2}=6^{2}\cdot(y^{3})^{2}\).
Calculating \(6^{2}=36\) and using the power of a power rule \((a^{m})^{n}=a^{mn}\), we get \((y^{3})^{2}=y^{3\times2}=y^{6}\). So, \((6y^{3})^{2}=36y^{6}\).

Step3: Multiply the simplified denominator by \(y\)

Now we have \(36y^{6}\cdot y\). Using the product of powers rule \(a^{m}\cdot a^{n}=a^{m + n}\), with \(a = y\), \(m = 6\), and \(n = 1\), we get \(y^{6+1}=y^{7}\). So the denominator becomes \(36y^{7}\).

Step4: Write the final expression

Putting it all together, our original expression simplifies to \(\frac{1}{36y^{7}}\).

Answer:

\(\boldsymbol{\frac{1}{36y^{7}}}\) (which corresponds to the last option: \(\frac{1}{36y^{7}}\))